`cot(v - u)` Find the exact value of the trigonometric expression given that sin(u) = -7/25 and cos(v) = -4/5 (Both u and v are in quadrant III.)

`sin(u)=-7/25`

using pythegorean identity,

`sin^2(u)+cos^2(u)=1`  

`(-7/25)^2+cos^2(u)=1`

`cos^2(u)=1-49/625=(625-49)/625=576/625`

`cos(u)=sqrt(576/625)=+-24/25`

Since u is in quadrant III , 

`:.cos(u)=-24/25`

`sin^2(v)+cos^2(v)=1`

`sin^2(v)+(-4/5)^2=1`

`sin^2(v)+16/25=1`

`sin^2(v)=1-16/25=(25-16)/25=9/25`

`sin(v)=sqrt(9/25)=+-3/5`

since v is in quadrant III,

`:.sin(v)=-3/5`

`cot(v-u)=cos(v-u)/sin(v-u)`

`cot(v-u)=(cos(v)cos(u)+sin(v)sin(u))/(sin(v)cos(u)-cos(v)sin(u))` 

plug in the values of sin(v),sin(u),cos(v) and cos(u),

`cot(v-u)=((-4/5*-24/25+(-3/5)*-7/25))/((-3/5*-24/25-(-4/5)*-7/25))`

`cot(v-u)=(96/125+21/125)/(72/125-28/125)`

`cot(v-u)=(117/125)/(44/125)`

`cot(v-u)=117/44`